| L(s) = 1 | − 15.1·2-s + 101.·4-s + 198.·7-s + 407.·8-s + 5.26e3·11-s + 1.21e3·13-s − 3.01e3·14-s − 1.91e4·16-s + 3.45e4·17-s + 1.86e4·19-s − 7.97e4·22-s + 3.33e4·23-s − 1.83e4·26-s + 2.01e4·28-s + 1.78e5·29-s − 2.37e5·31-s + 2.37e5·32-s − 5.23e5·34-s + 4.82e5·37-s − 2.81e5·38-s − 2.93e5·41-s + 4.43e5·43-s + 5.32e5·44-s − 5.04e5·46-s + 4.81e4·47-s − 7.83e5·49-s + 1.22e5·52-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 0.789·4-s + 0.219·7-s + 0.281·8-s + 1.19·11-s + 0.153·13-s − 0.293·14-s − 1.16·16-s + 1.70·17-s + 0.622·19-s − 1.59·22-s + 0.571·23-s − 0.204·26-s + 0.173·28-s + 1.35·29-s − 1.43·31-s + 1.27·32-s − 2.28·34-s + 1.56·37-s − 0.832·38-s − 0.665·41-s + 0.850·43-s + 0.942·44-s − 0.763·46-s + 0.0676·47-s − 0.951·49-s + 0.120·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.302434325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.302434325\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 15.1T + 128T^{2} \) |
| 7 | \( 1 - 198.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.26e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.21e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.45e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.86e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.33e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.78e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.37e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.82e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.93e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.43e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.81e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.66e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.75e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.15e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.29e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.71e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.67e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.44e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.71e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.52e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.44e7T + 8.07e13T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.79367423634741923074872192688, −9.726576668269865085186100977771, −9.183252855968205428904619130704, −8.094394556964716276239572043940, −7.34766378989276015983529444762, −6.16254006637822431023759158177, −4.71595784580111757360105760745, −3.28017547966459986873970273884, −1.55609807045255323780158099939, −0.812976919322838014316298995138,
0.812976919322838014316298995138, 1.55609807045255323780158099939, 3.28017547966459986873970273884, 4.71595784580111757360105760745, 6.16254006637822431023759158177, 7.34766378989276015983529444762, 8.094394556964716276239572043940, 9.183252855968205428904619130704, 9.726576668269865085186100977771, 10.79367423634741923074872192688
